Problem 10 - Calculus and its applications

Question

Ticket prices for NFL football games have experienced steady growth, as shown in the table below. $$ \begin{array}{c|c} \begin{array}{c} \text { NUMBER OF YEARS, } x, \\ \text { SINCE } 2009 \end{array} & \begin{array}{c} \text { AVERAGE TICKET PRICE } \\ \text { (dollars) } \end{array} \\ \hline 0 & \$ 73.18 \\ 1 & 76.47 \\ 2 & 77.36 \\ 3 & 79.09 \\ 4 & 81.54 \end{array} $$ a) Find the regression line, \(y=m x+b\). b) Use the regression line to predict the average ticket price for an NFL game in 2020 and 2025

Step-by-Step Solution

Answer

The regression line is \(y = 2.10x + 73.32\). The predicted ticket price for 2020 is \$96.42, and for 2025 is \$107.92.

Step 1: Identify Variables

We have two variables: the number of years since 2009, which we'll denote as \(x\), and the average ticket price, which we'll denote as \(y\). The data points given are \((0, 73.18), (1, 76.47), (2, 77.36), (3, 79.09), (4, 81.54)\).

Step 2: Calculate The Slope (m)

The formula for the slope \(m\) of a line is given by: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] where \(n\) is the number of data points. Plug in the sum values: \(\sum x = 10\), \(\sum y = 387.64\), \(\sum xy = 1183.75\), \(\sum x^2 = 30\), and \(n = 5\). Calculate \(m\).

Step 3: Calculate The Intercept (b)

The formula for the y-intercept \(b\) is: \[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2} \] Use the previously calculated sums to find the intercept \(b\).

Step 4: Write The Regression Line Equation

Substitute the calculated values of \(m\) and \(b\) into the equation \(y = mx + b\). This gives you the regression line equation.

Step 5: Predict 2020 Ticket Price

In 2020, \(x = 11\) since 2020 is 11 years after 2009. Substitute \(x = 11\) into the regression line equation to find \(y\), the predicted ticket price for 2020.

Step 6: Predict 2025 Ticket Price

In 2025, \(x = 16\) since 2025 is 16 years after 2009. Substitute \(x = 16\) into the regression line equation to find \(y\), the predicted ticket price for 2025.

Key Concepts

Linear Regression

Linear Regression is a fundamental statistical method used in data analysis to understand the relationship between two continuous variables. In our exercise, we're looking at the relationship between the year (measured as the number of years since 2009) and the average NFL ticket price. This is a classic case of predicting one variable based on another, a core application of linear regression.

The goal is to find the best-fit line through the data points. This line is typically expressed in the form of an equation: \( y = mx + b \). Here, \( y \) represents the dependent variable (average ticket price), \( x \) is the independent variable (years since 2009), \( m \) is the slope of the line, and \( b \) is the y-intercept.

• The slope \( m \) tells us how much the ticket price is expected to increase for each additional year.
• The intercept \( b \) is the predicted ticket price when \( x = 0 \), which corresponds to the year 2009.

With these parameters, you can predict future ticket prices if the conditions represent a continuation of the observed trend.

Slope Calculation

In order to accurately describe the relationship between years and ticket prices, calculating the slope \( m \) is crucial. The slope provides insight into the rate at which the ticket price changes according to the variable of time (number of years since 2009).

The formula for calculating the slope is: \[m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\]where:

• \( n \) is the number of data points.
• \( \sum xy \) is the sum of the products of corresponding \( x \) and \( y \) values.
• \( \sum x \) is the sum of all \( x \) values.
• \( \sum y \) is the sum of all \( y \) values.
• \( \sum x^2 \) is the sum of squares of all \( x \) values.

This equation helps derive \( m \), which reflects how much the cost of a ticket is expected to change as we move from one year to the next. A positive \( m \) indicates an upward trend, while a negative \( m \) would suggest a decrease.

Statistical Predictions

After determining the regression line, statistical predictions allow us to estimate average NFL ticket prices for future years. Once we have the regression equation, it's a straightforward process to make predictions by substituting values of \( x \) for the years of interest.

For instance, to predict the ticket price in 2020, we identify 2020 as 11 years after 2009, so \( x = 11 \). By substituting \( x = 11 \) into our regression equation, we can calculate the expected ticket price for that year. Similarly, for 2025, \( x = 16 \), and using this value in our regression equation allows us to predict the ticket price.

These predictions, while based on statistical methods, hinge on the assumption that the same growth pattern will continue into the future. Therefore, it's essential to apply these predictions with the understanding that real-world factors could alter these trends.

Data Analysis

Data Analysis is the backbone of linear regression, paving the way for meaningful insights and predictions. It involves not just crunching numbers but understanding the data context and patterns.

The process starts with collecting data—here, it's the average NFL ticket prices over a series of years. By meticulously analyzing each year and its corresponding ticket price, we can gain insights into broader trends.

• The key steps include compiling the data into a comprehensible format, such as a table or chart.
• Subsequently, essential statistical calculations help determine the central tendencies and variances.

This information becomes valuable to businesses, economists, and policymakers who may rely on such analyses to make informed decisions. They might use this data to evaluate if ticket price trends mirror economic conditions or if they need to strategize for changes in market dynamics. Data analysis, thus, translates raw numbers into actionable insights.